PASSIVITY is a property of engineering systems, used in a variety of
engineering disciplines, but most commonly found in analog electronics
and control systems . A PASSIVE COMPONENT, depending on field, may be
either a component that consumes (but does not produce) energy
(thermodynamic passivity), or a component that is incapable of power
gain (incremental passivity).

A component that is not passive is called an ACTIVE COMPONENT. An
electronic circuit consisting entirely of passive components is called
a passive circuit (and has the same properties as a passive
component). Used out-of-context and without a qualifier, the term
PASSIVE is ambiguous. Typically, analog designers use this term to
refer to INCREMENTALLY PASSIVE components and systems, while control
systems engineers will use this to refer to THERMODYNAMICALLY PASSIVE
ones.

Systems for which the small signal model is not passive are sometimes
called locally active (e.g. transistors and tunnel diodes). Systems
that can generate power about a time-variant unperturbed state are
often called parametrically active (e.g. certain types of nonlinear
capacitors).

In control systems and circuit network theory, a passive component or
circuit is one that consumes energy, but does not produce energy.
Under this methodology, voltage and current sources are considered
active, while resistors , capacitors , inductors , transistors ,
tunnel diodes , metamaterials and other dissipative and energy-neutral
components are considered passive. Circuit designers will sometimes
refer to this class of components as dissipative, or thermodynamically
passive.

While many books give definitions for passivity, many of these
contain subtle errors in how initial conditions are treated (and,
occasionally, the definitions do not generalize to all types of
nonlinear time-varying systems with memory). Below is a correct,
formal definition, taken from Wyatt et al. (which also explains the
problems with many other definitions). Given an n-port R with a state
representation S, and initial state x, define available energy EA as:
E A ( x ) = sup x T 0 0 T v ( t ) , i ( t )
d t {displaystyle E_{A}(x)=sup _{xto Tgeq 0}int _{0}^{T}-langle
v(t),i(t)rangle ,{mathord {operatorname {d} }}t}

where the notation supx→T≥0 indicates that the supremum is taken
over all T ≥ 0 and all admissible pairs {v(·), i(·)} with the
fixed initial state x (e.g., all voltage–current trajectories for a
given initial condition of the system). A system is considered passive
if EA is finite for all initial states x. Otherwise, the system is
considered active. Roughly speaking, the inner product v ( t ) ,
i ( t ) {displaystyle langle v(t),i(t)rangle } is the
instantaneous power (e.g., the product of voltage and current), and EA
is the upper bound on the integral of the instantaneous power (i.e.,
energy). This upper bound (taken over all T ≥ 0) is the available
energy in the system for the particular initial condition x. If, for
all possible initial states of the system, the energy available is
finite, then the system is called passive.

INCREMENTAL PASSIVITY

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In circuit design , informally, passive components refer to ones that
are not capable of power gain ; this means they cannot amplify
signals. Under this definition, passive components include capacitors
, inductors , resistors , diodes , transformers , voltage sources, and
current sources. They exclude devices like transistors , vacuum tubes
, relays , tunnel diodes, and glow tubes . Formally, for a memoryless
two-terminal element, this means that the current–voltage
characteristic is monotonically increasing . For this reason, control
systems and circuit network theorists refer to these devices as
locally passive, incrementally passive, increasing, monotone
increasing, or monotonic. It is not clear how this definition would be
formalized to multiport devices with memory – as a practical matter,
circuit designers use this term informally, so it may not be necessary
to formalize it.

This term is used colloquially in a number of other contexts:

* A passive USB to PS/2 adapter consists of wires, and potentially
resistors and similar passive (in both the incremental and
thermodynamic sense) components. An active USB to PS/2 adapter
consists of logic to translate signals (active in the incremental
sense)
* A passive mixer consists of just resistors (incrementally
passive), whereas an active mixer includes components capable of gain
(active).
* In audio work one can also find both (incrementally) passive and
active converters between balanced and unbalanced lines. A passive
bal/unbal converter is generally just a transformer along with, of
course, the requisite connectors, while an active one typically
consists of a differential drive or an instrumentation amplifier.

OTHER DEFINITIONS OF PASSIVITY

In some very informal settings, passivity may refer to the simplicity
of the device, although this definition is now almost universally
considered incorrect. Here, devices like diodes would be considered
active, and only very simple devices like capacitors, inductors, and
resistors are considered passive. In some cases, the term "linear
element " may be a more appropriate term than "passive device." In
other cases, "solid state device " may be a more appropriate term than
"active device."

STABILITY

Passivity, in most cases, can be used to demonstrate that passive
circuits will be stable under specific criteria. Note that this only
works if only one of the above definitions of passivity is used – if
components from the two are mixed, the systems may be unstable under
any criteria. In addition, passive circuits will not necessarily be
stable under all stability criteria. For instance, a resonant series
LC circuit will have unbounded voltage output for a bounded voltage
input, but will be stable in the sense of Lyapunov , and given bounded
energy input will have bounded energy output.

Passivity is frequently used in control systems to design stable
control systems or to show stability in control systems. This is
especially important in the design of large, complex control systems
(e.g. stability of airplanes). Passivity is also used in some areas of
circuit design, especially filter design.

PASSIVE FILTER

A passive filter is a kind of electronic filter that is made only
from passive components – in contrast to an active filter, it does
not require an external power source (beyond the signal). Since most
filters are linear, in most cases, passive filters are composed of
just the four basic linear elements – resistors, capacitors,
inductors, and transformers. More complex passive filters may involve
nonlinear elements, or more complex linear elements, such as
transmission lines. Television signal splitter consisting of a
passive high-pass filter (left) and a passive low-pass filter (right).
The antenna is connected to the screw terminals to the left of center.

A passive filter has several advantages over an active filter :

* Guaranteed stability
* Scale better to large signals (tens of amperes, hundreds of
volts), where active devices are often impractical
* No power supply needed
* Often less expensive in discrete designs (unless large coils are
required)
* For linear filters, potentially greater linearity depending on
components required

They are commonly used in speaker crossover design (due to the
moderately large voltages and currents, and the lack of easy access to
a power supply), filters in power distribution networks (due to the
large voltages and currents), power supply bypassing (due to low cost,
and in some cases, power requirements), as well as a variety of
discrete and home brew circuits (for low-cost and simplicity). Passive
filters are uncommon in monolithic integrated circuit design, where
active devices are inexpensive compared to resistors and capacitors,
and inductors are prohibitively expensive. Passive filters are still
found, however, in hybrid integrated circuits . Indeed, it may be the
desire to incorporate a passive filter that leads the designer to use
the hybrid format.

NOTES

* ^ This is probably formalized in one of the extensions to
Duffin's Theorem. One of the extensions may state that if the small
signal model is thermodynamically passive, under some conditions, the
overall system will be incrementally passive, and therefore, stable.
This needs to be verified.

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